Radial Limits of Capillary Surfaces at Corners

Abstract

Consider a solution $f\in C^{2}(\Omega)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ \Omega\subset R^{2}, \] where $\Omega$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partial\Omega$ and the angular measure of this corner is $2\alpha,$ for some $\alpha\in (0,\pi).$ Suppose $\sup_{x\in\Omega} |f(x)|$ and $\sup_{x\in\Omega} |H(x,f(x))|$ are both finite. If $\alpha>\frac{\pi}{2},$ then the (nontangential) radial limits of $f$ at ${\cal O},$ \[ Rf(\theta) = \lim_{r\downarrow 0} f(r\cos(\theta),r\sin(\theta)), \] were recently proven by the authors to exist, independent of the boundary behavior of $f$ on $\partial\Omega,$ and to have a specific type of behavior. Suppose $\alpha\in \left(\frac{\pi}{4},\frac{\pi}{2}\right],$ the contact angle $\gamma(\cdot)$ that the graph of $f$ makes with one side of $\partial\Omega$ has a limit (denoted $\gamma_{2}$) at ${\cal O}$ and \[ \pi-2\alpha < \gamma_{2} <2\alpha. \] We prove that the (nontangential) radial limits of $f$ at ${\cal O}$ exist and the radial limits have a specific type of behavior, independent of the boundary behavior of $f$ on the other side of $\partial\Omega.$ We also discuss the case $\alpha\in \left(0,\frac{\pi}{2}\right].$